Question

In \(\Delta\)ABC, P is a point on AB and Q is a point on AC such that PQ \(\parallel\) BC. If AP : PB = 3 : 2, then PQ : BC is equal to :

• A. 3 : 2
• B. 2 : 5
• C. 3 : 5
• D. 5 : 3

Hint:

Always remember that when a line is drawn parallel to one side of a triangle, the smaller triangle formed is similar to the original larger triangle.
A common mistake is to equate \(\frac{PQ}{BC}\) to \(\frac{AP}{PB}\). This is incorrect!
The parallel segment ratio always corresponds to the ratio of the side of the small triangle to the side of the large triangle, which is:
\[ \frac{\text{Small Side}}{\text{Large Side}} = \frac{AP}{AB} = \frac{AP}{AP + PB} \]

Answer 1

The Correct Option is C

Step 1: Understand what is being asked.
In $\Delta ABC$, $P$ lies on $AB$, $Q$ lies on $AC$, and $PQ \parallel BC$. Given $AP:PB=3:2$, we need $PQ:BC$. Let us work with the whole side $AB$ built from the given ratio, and use the direct corollary of the Basic Proportionality Theorem for parallel sides in similar triangles.

Step 2: Express $AP$ and $AB$ in terms of a single variable.
Since $AP:PB=3:2$, let $AP=3m$ and $PB=2m$ for some length $m$. Then the whole side is:
\[ AB = AP+PB = 3m+2m = 5m \]
Step 3: Establish similarity of the triangles.
Since $PQ \parallel BC$, the line $PQ$ cuts off a smaller triangle $APQ$ inside $\Delta ABC$ that shares the angle at $A$, and has equal corresponding angles at $P$ and $Q$ (corresponding angles with the parallel lines). By AA similarity:
\[ \Delta APQ \sim \Delta ABC \]
Step 4: Use the similarity ratio, matching $A$ to $A$, $P$ to $B$, and $Q$ to $C$.
In similar triangles, every pair of corresponding sides has the same ratio, so the side $PQ$ (facing vertex $A$ in the small triangle) compares to side $BC$ (facing vertex $A$ in the big triangle) exactly as side $AP$ compares to side $AB$:
\[ \frac{PQ}{BC} = \frac{AP}{AB} = \frac{3m}{5m} = \frac{3}{5} \]
Final Answer:
The ratio $PQ:BC$ is $3:5$, matching option (C).
\[ \boxed{3:5} \]